Definitions Flashcards
Starlike
A domain is called starlike if there exists a point in D such that for any other point in D there is a straight line connecting a and b entirely inside D
Biholomorphic
Let D and Dā be domains. We say that f:D -> Dā is biholomorphic if f is holomorphic and a bijection, the inverse of which is also holomorphic.
Domain
An open path connected set.
Metric Space
A set with a function satisfying positivity, symmetry and the triangle inequality.
Meromorphic
A function is meromorphic on a domain D if f is holomorphic on D-S where S has no isolated points and f has a pole at each element of S.
Principal Part
The negative section of a laurent series.
Simple Closed Curve
A closed curve is called simple if f(t1) = f(t2) for any t1
Simply connected
An open set is called simply connected if every closed contour is homologous to 0 i.e. winding number 0.
Holomorphic on an Open set
A complex function is holomorphic on an open set if it is complex differentiable at every point in the open set.
Holomorphic at a point.
A complex function is holomorphic at a point in an open set if it is holomorphic on some open ball around that point.
Conformal
We say a real differentiable map on a domain is conformal at a point if it preserves the angle and orientation between any two tangent vectors at that point.
Uniform Convergence
A sequence of functions converges uniformly if for every epsilon greater than zero there exists an N in the natural numbers such that for every n>N d(fn(x),f(x))
Complex Differentiable
A function is complex differentiable at a point a in the open set if the limit from z to a of f(z)-f(a)/z-a exists.
Non isolated point
Given a subset S of C we say that a point w in S is a non isolated point if for every epsilon greater than 0 there exists a z in S such that the 0
Harmonic Function
A real valued function is harmonic on a domain if it has continuous second order partial differentials and Uxx+Uyy = 0
Local Uniform Convergence
A sequence of functions converges locally uniformly on X to f if for every x in X there exists an open set U in X such that x is in U and fn converges to f uniformly.
Length of a contour
The integral between the end points of the modulus of the differential of the contour.
Compactness
A non empty subset K of a metric space X is called compact if for any sequence in K there exists a convergent subsequence with limit in K.
Open Set
A subset is open if for every x in the subset there exists an epsilon such that the ball of radius epsilon around x is contained in the subset.
Closed Set
A subset is closed if its complement is open.
Convergence in a metric space
A sequence is convergent in a metric space if for every epsilon greater than zero there exists an N in the natural numbers such that d(xn,x)N
Continuity
For every epsilon greater than 0 there exists a delta greater than 0 such that for every x in X1, d(x,x0)
Pointwise convergent
For every x in X and for every epsilon>0 there exists an N in the natural numbers such that for every n >N d(fn(x),f(x))
